That page lists all the sequences featuring 0, 1, 3, 8, 15 that occur in a very comprehensive list of sequences occurring in mathematics. None of them is given by a simple formula, and there does not seem to a natural formula that generates your sequence.

If I had to guess, I would say that the "1" should not be in the sequence that you were given, and that your suggested answer $\displaystyle n^2-1$ (where $\displaystyle n\geqslant1$) is the one that was expected.